A Darboux theorem for Hamiltonian operators in the formal calculus of variations

TL;DR

This paper establishes a Darboux theorem for formal deformations of Hamiltonian operators of hydrodynamic type, revealing a moduli space of normal forms and employing advanced algebraic structures to describe deformation equivalences.

Contribution

It introduces a Darboux theorem for Hamiltonian operators in the formal calculus of variations, highlighting the structure of their deformation space and utilizing dg Lie algebras and 2-groupoids.

Findings

Not all deformations are equivalent to the original operator.

There exists a moduli 2-stack of normal forms for deformations.

The paper employs a refined Schouten bracket in the formal calculus of variations.

Abstract

We prove a Darboux theorem for formal deformations of Hamiltonian operators of hydrodynamic type (Dubrovin-Novikov). Not all deformations are equivalent to the original operator: there is a moduli 2-stack of normal forms. The paper utilizes three main concepts: 1) dg Lie algebras concentrated in degrees [-1,\infty) such as the Schouten algebra - these give a convenient language for describing deformation problems; 2) the Deligne 2-groupoid associated to such a dg Lie algebra, which represents the moduli of formal deformations; 3) a refined version of the Schouten bracket in the formal calculus of variations, due to V. O. Soloviev (hep-th/9305133).